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\newcommand{\g}{\mathfrak{g}} \newcommand{\sfr}{\mathfrak{s}} \newcommand{\zfr}{\mathfrak{z}} \newcommand{\ufr}{\mathfrak{u}} \newcommand{\afr}{\mathfrak{a}} \newcommand{\so}{\mathfrak{so}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calL}{\mathcal{L}} \newcommand{\schouten}[2]{{#1,#2}}

\pdfstringdefDisableCommands{% \def\g{g}% \def\sfr{s}% \def\zfr{z}% \def\ufr{u}% \def\afr{a}% \def\calM{M}% \def\calR{R}% \def\calC{C}% \def\Ob{Ob}% }

\title[Hochschild--Serre $J$-channel obstructions]{Hochschild--Serre \texorpdfstring{$J$}{J}-Channel Obstructions for Flat Metric Lie Bialgebras} \author{Amine Bahayou} \address{Department of Mathematics, Kasdi Merbah University, Ouargla 30000, Algeria} \email{[email protected]} \subjclass[2020]{Primary 17B62, 53D17; Secondary 17B56, 18G40, 53D30, 81R50, 81R60} \keywords{Lie bialgebra; flat metric Lie algebra; Milnor decomposition; Hochschild--Serre spectral sequence; Poisson--Lie group; deformation quantization; resonant degeneration; $\kappa$-Poincar'e} \date{\today}

\begin{document}

\begin{abstract} Let

$$\g=\sfr\oplus\zfr\oplus\ufr, \qquad \ufr=[\g,\g]=\bigoplus_{\ell=1}^{m}P_{\ell}, \qquad \ad_s|_{P_{\ell}}=\lambda_{\ell}(s)J_{\ell},$$

be a real flat metric Lie algebra in Milnor form. We study normalized Lie bialgebra one-cocycles whose restriction to a rotation plane contains, in addition to the identity-type term $A_{\ell}\wedge u$, a central rotated term $W_{\ell}\wedge J_{\ell}u$. The central rotated sector produces a Hochschild--Serre class

$$[\sigma_R]\in H^0\bigl(\afr,H^1(\ufr,\wedge^3\g)\bigr), \qquad \sigma_R(u)=J_{\ell}u\wedge R(W_{\ell})\quad (u\in P_{\ell}),$$

whose comparison image is an explicit cokernel obstruction $\Ob(R)\in\coker\beta$. The main result proves that the $J$-extended coupled Maurer--Cartan system is solvable if and only if this class vanishes. The proof uses actual coefficient cohomology for the abelian ideal $\ufr$, not the erroneous identification $H^1(\ufr,M)=\Hom(\ufr,M)$.

The revision gives a cochain-level definition of strict $\afr$-invariance and separates it from invariance only in cohomology, proves rank and surjectivity criteria for the contraction map $\beta$, supplies a projector-based canonicity proof, expands the polynomial integration argument using right trivialization and the exact BCH truncation on $N=\exp(\zfr\oplus\ufr)$, and states a functional-analytic version of the simple-pole theorem at resonance. The examples include a no-go result for genuine four-dimensional obstructions and a fully worked example with four-dimensional nilradical exhibiting both $\Ob(R)\neq0$ and $\Ob(R)=0$. \end{abstract}

\maketitle \tableofcontents

\section*{Notation and standing conventions}\label{sec:notation} \addcontentsline{toc}{section}{Notation and standing conventions}

All spaces are real unless stated otherwise. The table fixes the notation used throughout. \begin{center} \renewcommand{\arraystretch}{1.18} \begin{tabular}{p{0.22\textwidth}p{0.67\textwidth}} \hline Symbol & Meaning \ \hline $\g$ & flat metric Lie algebra. \ $\sfr$ & abelian Milnor subalgebra acting by rotations on $\ufr$. \ $\zfr$ & centre $Z(\g)$. \ $\ufr$ & derived ideal $[\g,\g]$, abelian in Milnor's class. \ $\afr$ & quotient subalgebra $\sfr\oplus\zfr$, so $\g=\afr\ltimes\ufr$. \ $P_{\ell}$ & oriented two-plane in $\ufr$. \ $J_{\ell}$ & positive quarter-turn on $P_{\ell}$, $J_{\ell}^{2}=-\id$. \ $\lambda_{\ell}$ & weight $\ad_s|_{P_{\ell}}=\lambda_{\ell}(s)J_{\ell}$. \ $\omega_{\ell}$ & oriented area bivector of $P_{\ell}$. \ $A_{\ell}$ & identity-type coefficient. \ $W_{\ell}$ & central rotated coefficient in $\zfr$. \ $I^2$ & invariant bivectors $(\wedge^2\g)^\g$. \ $T_{\ell}$ & visible quotient $I^2/\R\omega_{\ell}$. \ $\beta_{\ell}$ & contraction $(\wedge^3\g)^\afr\to T_{\ell}$. \ $\calM_{ij}^{\sigma}$ & mixed resonant plane of weight $\lambda_i+\sigma\lambda_j$. \ $\sigma_R$ & cochain $u\mapsto J_{\ell}u\wedge R(W_{\ell})$. \ $\Ob(R)$ & class of $(\pi_{\ell}R(W_{\ell}))_{\ell}$ in $\coker\beta$. \ \hline \end{tabular} \end{center}

\section{Introduction}

Milnor's structure theorem for flat left-invariant metrics \cite{Milnor1976} gives an orthogonal decomposition \begin{equation}\label{eq:milnor-intro} \g=\sfr\oplus\zfr\oplus\ufr, \qquad \zfr=Z(\g), \qquad \ufr=[\g,\g], \end{equation} where $\sfr$ and $\zfr$ are abelian, $\ufr$ is an abelian ideal, and each $\ad_s|_{\ufr}$ is skew-adjoint. Simultaneous real block diagonalization gives \begin{equation}\label{eq:weights-intro} \ufr=\bigoplus_{\ell=1}^{m}P_{\ell}, \qquad \ad_s|{P{\ell}}=\lambda_{\ell}(s)J_{\ell}. \end{equation} This architecture reduces many Lie bialgebra questions to finite-dimensional weight linear algebra.

Lie bialgebras are infinitesimal multiplicative Poisson structures \cite{Drinfeld1983,KosmannSchwarzbach2004}. Their quantization is governed by deformation-theoretic machinery in the sense of Drinfeld and Etingof--Kazhdan \cite{Drinfeld1987,EtingofKazhdan1996}. Low-dimensional solvable and Euclidean examples, especially $\mathfrak e(2)$, have been classified in explicit form in \cite{Gomez2000,Stachura1998,Zakrzewski1997}. The present paper isolates a cohomological mechanism that is not visible in the purely identity-type normalization: the central rotated $J$-channel. The Hochschild--Serre spectral sequence is used here as an obstruction mechanism in the spirit of \cite{HochschildSerre1953,Gerstenhaber1964,NijenhuisRichardson1967,Weibel1994}. The singular resonant part is motivated by degeneration phenomena familiar in $\kappa$-Poincar'e and related quantum-group limits \cite{MajidRuegg1994,LukierskiRueggZakrzewski1995,BorowiecPachol2010}.

The main theorem is the following.

\begin{theorem}[Central rotated Hochschild--Serre obstruction]\label{thm:intro-main} Let $R$ be a $J$-extended normalized one-cocycle on a nonresonant flat metric Lie algebra $\g$. The cochain $\sigma_R$ defines a strict $\afr$-invariant class

$$[\sigma_R]\in H^0\bigl(\afr,H^1(\ufr,\wedge^3\g)\bigr).$$

Under the comparison isomorphism

$$\Gamma:H^0\bigl(\afr,H^1(\ufr,\wedge^3\g)\bigr)_J\xrightarrow{\sim}\coker\beta$$

one has $\Gamma([\sigma_R])=\Ob(R)$. The $J$-extended coupled Maurer--Cartan system is solvable if and only if $\Ob(R)=0$. \end{theorem}

The proof appears as \Cref{thm:obstruction}. The strictness hypothesis in the survival statement is sufficient, not logically necessary: a class invariant only in cohomology survives exactly when its Hochschild--Serre transgression vanishes; this is made precise in \Cref{prop:cohomological-invariance}.

\section{Milnor architecture and invariant tensors}\label{sec:foundations}

\begin{definition}[Flat metric Lie algebra] A flat metric Lie algebra is a pair $(\g,\langle\cdot,\cdot\rangle)$ such that the simply connected Lie group with Lie algebra $\g$ and the induced left-invariant metric has zero sectional curvature. \end{definition}

\begin{theorem}[Milnor decomposition \cite{Milnor1976}]\label{thm:milnor} Let $(\g,\langle\cdot,\cdot\rangle)$ be a flat metric Lie algebra. Then there is an orthogonal decomposition

$$\g=\sfr\oplus\zfr\oplus\ufr, \qquad \zfr=Z(\g), \qquad \ufr=[\g,\g],$$

in which $\sfr$ and $\zfr$ are abelian, $\ufr$ is an abelian ideal, and $\ad_s|_{\ufr}\in\so(\ufr)$ for every $s\in\sfr$. The commuting skew-adjoint operators $\ad_s|_{\ufr}$ admit a simultaneous decomposition

$$\ufr=\bigoplus_{\ell=1}^{m}P_{\ell}, \qquad \ad_s|_{P_{\ell}}=\lambda_{\ell}(s)J_{\ell},$$

where $P_{\ell}$ is an oriented Euclidean two-plane, $J_{\ell}^{2}=-\id$, and $\lambda_{\ell}\in\sfr^{\ast}\setminus\{0\}$. \end{theorem}

We put $\afr=\sfr\oplus\zfr$ and extend $\lambda_{\ell}$ by zero on $\zfr$. Choose oriented orthonormal bases $(e_{\ell},f_{\ell})$ of $P_{\ell}$ with $J_{\ell}e_{\ell}=f_{\ell}$ and write $\omega_{\ell}=e_{\ell}\wedge f_{\ell}$.

\begin{definition}[Nonresonance]\label{def:nonresonant} The weight system is nonresonant if: \begin{enumerate}[label=\textup{(N\arabic*)}] \item\label{item:N1} $\lambda_i\neq\pm\lambda_j$ for $i\neq j$; \item\label{item:N2} for pairwise distinct $i,j,k$ and $a,b\in\{-1,0,1\}$, $\lambda_k=a\lambda_i+b\lambda_j$ never holds. \end{enumerate} \end{definition}

\begin{proposition}[Invariant tensors]\label{prop:invariant-tensors} Under \Cref{def:nonresonant}, \begin{align} (\wedge^2\g)^\g&=\wedge^2\zfr\oplus\bigoplus_{\ell=1}^{m}\R\omega_{\ell},\label{eq:inv2}\ (\wedge^3\g)^\g&=\wedge^3\zfr\oplus\bigoplus_{\ell=1}^{m}\zfr\wedge\R\omega_{\ell}\oplus\bigoplus_{\ell=1}^{m}\sfr_{\ell}\wedge\R\omega_{\ell},\label{eq:inv3} \end{align} where $\sfr_{\ell}=\bigcap_{k\neq\ell}\ker\lambda_k$. Moreover \begin{equation}\label{eq:ainv3} (\wedge^3\g)^\afr=\wedge^3\afr\oplus\bigoplus_{\ell=1}^{m}\afr\wedge\R\omega_{\ell}. \end{equation} \end{proposition}

\begin{proof} Decompose exterior powers into $\sfr$-weight spaces. The zero-weight part of $\wedge^2\g$ is $\wedge^2\afr\oplus\bigoplus\R\omega_{\ell}$ by \ref{item:N1}. Invariance under $\ufr$ removes every noncentral $\sfr$-factor in $\wedge^2\afr$, leaving \eqref{eq:inv2}. In degree three, \ref{item:N2} excludes zero-weight triples from three distinct planes; the $\afr$-invariant part is therefore \eqref{eq:ainv3}. Requiring also $\ufr$-invariance removes $\sfr$-factors except those annihilating all other weights, giving \eqref{eq:inv3}. \end{proof}

\section{The coupled Maurer--Cartan system}\label{sec:coupled}

A Lie bialgebra cobracket is a Chevalley--Eilenberg one-cocycle $\xi\in Z^1(\g,\wedge^2\g)$ satisfying the co-Jacobi equation $\schouten{\xi}{\xi}=0$. In Milnor normal form one writes $\xi=\ad_r+R$ and \begin{equation}\label{eq:MC-expansion} \schouten{\xi}{\xi}=\schouten{R}{R}+d_{\g}\Psi, \qquad \Psi=[r,r]+2\schouten{r}{R}. \end{equation}

\begin{definition}[$J$-extended normalized cocycle]\label{def:Jextended} A normalized cocycle $R\in Z^1(\g,\wedge^2\g)$ is $J$-extended if \begin{equation}\label{eq:Jextended} R(u)=A_{\ell}\wedge u+W_{\ell}\wedge J_{\ell}u, \qquad u\in P_{\ell}, \end{equation} where $A_{\ell}\in\afr$, $W_{\ell}\in\zfr$, and $R(\afr)\subset(\wedge^2\g)^\g$. The second term is the central rotated, or $J$-channel, term. \end{definition}

Let

$$I^2=(\wedge^2\g)^\g=\wedge^2\zfr\oplus\bigoplus_{k=1}^{m}\R\omega_k, \qquad T_{\ell}=I^2/\R\omega_{\ell}, \qquad \calB_J=\bigoplus_{\ell=1}^{m}T_{\ell},$$

and write $\pi_{\ell}:I^2\to T_{\ell}$ for the quotient.

\begin{definition}[Canonical contraction]\label{def:beta} For $\Psi\in(\wedge^3\g)^\afr$, define $\beta_{\ell}(\Psi)\in T_{\ell}$ by \begin{equation}\label{eq:beta-def} [u,\Psi]\equiv -J_{\ell}u\wedge\beta_{\ell}(\Psi) \quad\bmod J_{\ell}u\wedge\R\omega_{\ell}, \qquad u\in P_{\ell}. \end{equation} Set $\beta=\bigoplus_{\ell}\beta_{\ell}:(\wedge^3\g)^\afr\to\calB_J$. \end{definition}

\begin{proposition}[Coordinate formula]\label{prop:beta-coordinate} Write

$$\Psi=\alpha+\sum_{k=1}^{m}a_k\wedge\omega_k, \qquad \alpha\in\wedge^3\afr, \quad a_k\in\afr,$$

as in \eqref{eq:ainv3}. Then \begin{equation}\label{eq:beta-coordinate} \beta_{\ell}(\Psi)=\pi_{\ell}\left(\pr_{I^2}\iota_{\lambda_{\ell}}\alpha+ \sum_{k\neq\ell}\lambda_{\ell}(a_k)\omega_k\right). \end{equation} \end{proposition}

\begin{proof} Only brackets with $\sfr$-factors contribute. Since $[u,s]=-\lambda_{\ell}(s)J_{\ell}u$ and $[u,\zfr]=0$, contraction of one $\sfr$-factor in $\alpha$ gives the first term. Similarly, $[u,a_k\wedge\omega_k]=-\lambda_{\ell}(a_k)J_{\ell}u\wedge\omega_k$ for $k\neq\ell$. The $k=\ell$ contribution is invisible because $J_{\ell}u\wedge\omega_{\ell}=0$. \end{proof}

\begin{proposition}[Rank bounds and surjectivity]\label{prop:rank} Let $p=\dim\sfr$, $q=\dim\zfr$, and $a=p+q$. Under nonresonance, \begin{equation}\label{eq:rank-dimensions} \dim(\wedge^3\g)^\afr=\binom{a}{3}+am, \qquad \dim\calB_J=m\left(\binom{q}{2}+m-1\right). \end{equation} Consequently

$$\rk\beta\leq\min\left\{\binom{a}{3}+am, m\left(\binom{q}{2}+m-1\right)\right\}, \qquad \dim\coker\beta=\dim\calB_J-\rk\beta.$$

Each component $\beta_{\ell}$ is onto $T_{\ell}$. The total map $\beta$ is onto if the evaluation map \begin{equation}\label{eq:evaluation} L:\sfr\to\R^m, \qquad L(s)=(\lambda_1(s),\ldots,\lambda_m(s)), \end{equation} is onto. The surjectivity of $\beta$ is independent of the identity-type coefficients $A_{\ell}$; if one restricts the ansatz to trivectors of the form $A_k\wedge\omega_k$, the corresponding restricted matrix is $(\lambda_{\ell}(A_k))_{\ell\neq k}$. \end{proposition}

\begin{proof} The dimension formulas follow from \eqref{eq:ainv3} and from the definition of $T_{\ell}$. Since $\lambda_{\ell}\neq0$, choose $s_{\ell}$ with $\lambda_{\ell}(s_{\ell})=1$. For $\eta\in\wedge^2\zfr$, the trivector $s_{\ell}\wedge\eta$ maps to $\eta$ in $T_{\ell}$; for $k\neq\ell$, the trivector $s_{\ell}\wedge\omega_k$ maps to $\omega_k$. Thus $\beta_{\ell}$ is surjective. Simultaneous surjectivity requires solving the evaluations for all components at once. This is guaranteed by the surjectivity of $L$, which gives arbitrary prescribed values for all weights. \end{proof}

\begin{definition}[Central rotated obstruction]\label{def:obstruction} For a $J$-extended normalized cocycle $R$, define \begin{equation}\label{eq:obstruction} \Ob(R)=\left[(\pi_{\ell}R(W_{\ell}))_{\ell=1}^{m}\right]\in\coker\beta. \end{equation} \end{definition}

\begin{theorem}[Coupled Maurer--Cartan system]\label{thm:coupled} Let $R$ be $J$-extended and nonresonant. For $\xi=\ad_r+R$ and $\Psi=[r,r]+2\schouten{r}{R}$, the co-Jacobi equation is equivalent to: \begin{enumerate}[label=\textup{(C\arabic*)}] \item\label{item:C1} the reduced identity-type equation $\schouten{R}{R}=0$ on the non-$J$ blocks; \item\label{item:C2} $\Psi\in(\wedge^3\g)^\afr$; \item\label{item:C3} $(\pi_{\ell}R(W_{\ell}))_{\ell}=\beta(\Psi)$ in $\calB_J$. \end{enumerate} In particular, the $J$-channel is solvable for some $r$ precisely when $\Ob(R)=0$. \end{theorem}

\begin{proof} Equation \eqref{eq:MC-expansion} splits into $\sfr$-isotypic components. Nonresonance prevents cancellation among distinct channels. The identity-type part gives \ref{item:C1}; the zero-weight part of the coboundary equation gives \ref{item:C2}; and the coefficient of $J_{\ell}u$ modulo the invisible line $\R\omega_{\ell}$ is exactly \ref{item:C3} by \eqref{eq:beta-def}. Conversely these conditions annihilate every isotypic component of \eqref{eq:MC-expansion}. \end{proof}

\begin{proposition}[Mild resonances violating only \ref{item:N2}]\label{prop:mild} Assume \ref{item:N1} holds but \ref{item:N2} fails. Then \ref{item:C1}--\ref{item:C3} remain sufficient only after imposing vanishing of all resonant mixed components. In general the complete $J$-channel equation is \begin{equation}\label{eq:mild-C3} (\pi_{\ell}R(W_{\ell})){\ell}=\beta(\Psi_0)+\beta^{\mathrm{res}}(\Psi{\mathrm{res}}), \end{equation} where $\Psi_0$ is the nonresonant part of \eqref{eq:ainv3} and $\Psi_{\mathrm{res}}$ is a sum of zero-weight terms \begin{equation}\label{eq:mild-terms} P_i^{\epsilon_i}\wedge P_j^{\epsilon_j}\wedge P_k^{\epsilon_k}, \qquad \epsilon_i\lambda_i+\epsilon_j\lambda_j+ \epsilon_k\lambda_k=0, \qquad \epsilon_r\in{\pm1}. \end{equation} Here $\beta^{\mathrm{res}}$ is defined by the same contraction rule as \eqref{eq:beta-def}. Thus the original conditions are necessary and sufficient under a mild \ref{item:N2}-resonance exactly on the sublocus where $\beta^{\mathrm{res}}(\Psi_{\mathrm{res}})=0$ and the resonant component of $\schouten{R}{R}+d_{\g}\Psi$ vanishes. \end{proposition}

\begin{proof} When \ref{item:N1} holds, different rotation planes are still separated, so there is no multiplicity in the basic $J_{\ell}$-channel. The failure of \ref{item:N2} creates additional zero-weight trivectors from three distinct planes. Bracketing such a term with $u\in P_{\ell}$ may leave a visible coefficient in $T_{\ell}$, and these coefficients are precisely the terms collected in $\beta^{\mathrm{res}}$. No other weights can enter the $J$-channel because all nonzero weights remain in distinct isotypic components. \end{proof}

\section{The Hochschild--Serre obstruction}\label{sec:HS}

Let $M=\wedge^3\g$. For the split extension

$$0\to\ufr\to\g\to\afr\to0$$

the Hochschild--Serre spectral sequence is \begin{equation}\label{eq:HS} E_2^{p,q}=H^p\bigl(\afr,H^q(\ufr,M)\bigr)\Longrightarrow H^{p+q}(\g,M). \end{equation} Although $\ufr$ is abelian, its coefficient action on $M$ is not trivial.

\begin{lemma}[Abelian ideal cohomology]\label{lem:abelian} For an abelian Lie algebra $\ufr$ acting on $M$,

$$H^1(\ufr,M)= \frac{\{\phi:\ufr\to M:\ u\cdot\phi(v)=v\cdot\phi(u)\ \forall u,v\}} {\{u\mapsto u\cdot\Theta:\Theta\in M\}}.$$

\end{lemma}

\begin{definition}[Strict and cohomological invariance]\label{def:strict} A $d_{\ufr}$-closed cochain $\phi\in C^1(\ufr,M)$ is strictly $\afr$-invariant if \begin{equation}\label{eq:strict} a\cdot\phi(u)-\phi([a,u])=0 \qquad (a\in\afr, \ u\in\ufr). \end{equation} It is $\afr$-invariant in cohomology if for each $a\in\afr$ there exists $\Theta_a\in M$ such that \begin{equation}\label{eq:cohinv} a\cdot\phi(u)-\phi([a,u])=u\cdot\Theta_a. \end{equation} Strict invariance is a cochain-level statement; cohomological invariance is the statement that $[\phi]\in H^1(\ufr,M)$ belongs to the $\afr$-invariant subspace. \end{definition}

\begin{definition}[The obstruction cochain]\label{def:sigma} For a $J$-extended cocycle $R$, put \begin{equation}\label{eq:sigma} \sigma_R(u)=J_{\ell}u\wedge R(W_{\ell}), \qquad u\in P_{\ell}. \end{equation} \end{definition}

\begin{lemma}[Closedness and strict invariance]\label{lem:closed} The cochain $\sigma_R$ is $d_{\ufr}$-closed and strictly $\afr$-invariant. Hence it defines a class

$$[\sigma_R]\in H^0\bigl(\afr,H^1(\ufr,\wedge^3\g)\bigr).$$

\end{lemma}

\begin{proof} Put $B_{\ell}=R(W_{\ell})$. Since $W_{\ell}\in\zfr\subset\afr$ and $R(\afr)\subset(\wedge^2\g)^\g$, each $B_{\ell}$ is $\g$-invariant. For $u\in P_i$, $v\in P_j$,

$$(d_{\ufr}\sigma_R)(u,v)=u\cdot(J_jv\wedge B_j)-v\cdot(J_iu\wedge B_i)=0,$$

because $\ufr$ is abelian and $u\cdot B_j=v\cdot B_i=0$. For $s\in\sfr$ and $u\in P_{\ell}$,

$$[s,J_{\ell}u\wedge B_{\ell}]-\sigma_R([s,u]) =-\lambda_{\ell}(s)u\wedge B_{\ell}-\lambda_{\ell}(s)J_{\ell}^2u\wedge B_{\ell}=0.$$

Central elements act trivially, so \eqref{eq:strict} holds. \end{proof}

\begin{definition}[$J$-channel subspace]\label{def:Jsubspace} A $J$-channel cochain has the form $\phi(u)=J_{\ell}u\wedge B_{\ell}$ for $u\in P_{\ell}$ and $B_{\ell}\in I^2$. The corresponding subspace of $H^0(\afr,H^1(\ufr,M))$ is denoted $E_{2,J}^{0,1}$. \end{definition}

\begin{lemma}[Comparison isomorphism]\label{lem:comparison} The map

$$\Gamma:E_{2,J}^{0,1}\to\coker\beta, \qquad [\phi]\mapsto[(\pi_{\ell}B_{\ell})_{\ell}], \quad \phi(u)=J_{\ell}u\wedge B_{\ell},$$

is a well-defined isomorphism. Moreover $\Gamma([\sigma_R])=\Ob(R)$. \end{lemma}

\begin{proof} The component $B_{\ell}\in\R\omega_{\ell}$ is invisible because $J_{\ell}u\wedge\omega_{\ell}=0$. Coboundaries in the $J$-channel are exactly cochains $u\mapsto[u,\Theta]$ with $\Theta\in(\wedge^3\g)^\afr$, and \eqref{eq:beta-def} identifies their projected coefficients with $\beta(\Theta)$. Thus quotienting by $d_{\ufr}$-boundaries is exactly quotienting $\calB_J$ by $\im\beta$. The identity for $\sigma_R$ is \eqref{eq:obstruction}. \end{proof}

\begin{lemma}[Strict extension]\label{lem:extension} If $\phi\in C^1(\ufr,M)$ is $d_{\ufr}$-closed and strictly $\afr$-invariant, then the extension $\widetilde\phi|_{\ufr}=\phi$, $\widetilde\phi|_{\afr}=0$ is a one-cocycle in $C^1(\g,M)$. \end{lemma}

\begin{proof} The only nontrivial mixed differential is $(d_{\g}\widetilde\phi)(a,u)=a\cdot\phi(u)-\phi([a,u])$, which vanishes by strict invariance. The $\ufr\wedge\ufr$ component is $d_{\ufr}\phi$ and the $\afr\wedge\afr$ component is zero. \end{proof}

\begin{proposition}[Survival and cohomological invariance]\label{prop:cohomological-invariance} Every strict $\afr$-invariant $J$-channel class survives from $E_2^{0,1}$ to $E_{\infty}^{0,1}$. If a class is only invariant in cohomology, then it survives if and only if its Hochschild--Serre transgression $d_2[\phi]\in H^2(\afr,M^{\ufr})$ vanishes. \end{proposition}

\begin{proof} Strict classes extend to total one-cocycles by \Cref{lem:extension}; the transgression is the obstruction to such an extension, hence $d_2=0$ on them. For a cohomologically invariant representative satisfying \eqref{eq:cohinv}, one kills the $(1,1)$ component of $d\widetilde\phi$ by choosing primitives $\Theta_a$. The remaining $(2,0)$ defect is the standard Hochschild--Serre transgression. Since no higher differential can leave bidegree $(0,1)$ in a first-quadrant spectral sequence, vanishing of $d_2$ is equivalent to survival. \end{proof}

\begin{theorem}[Obstruction theorem]\label{thm:obstruction} Let $R$ be a $J$-extended normalized cocycle on a nonresonant flat metric Lie algebra. Then $[\sigma_R]\in E_{2,J}^{0,1}$ survives to $E_{\infty,J}^{0,1}$, and

$$\Gamma([\sigma_R])=\Ob(R).$$

The coupled Maurer--Cartan system of \Cref{thm:coupled} is solvable if and only if $[\sigma_R]=0$ in $E_{\infty,J}^{0,1}$, equivalently if and only if $\Ob(R)=0$. \end{theorem}

\begin{proof} Closedness and strict invariance are \Cref{lem:closed}; survival is \Cref{prop:cohomological-invariance}; the comparison with $\coker\beta$ is \Cref{lem:comparison}; and solvability is the vanishing criterion in \Cref{thm:coupled}. \end{proof}

\subsection{Canonicity}

Let

$$\wedge^2\afr=\wedge^2\sfr\oplus(\sfr\wedge\zfr)\oplus\wedge^2\zfr$$

with projectors $\Pi_{\sfr\sfr}$, $\Pi_{\sfr\zfr}$, and $\Pi_{\zfr\zfr}$.

\begin{lemma}[Projector separation]\label{lem:projectors} Let $q\in\wedge^2\afr$ and write $q=q_{\sfr\sfr}+q_{\sfr\zfr}+q_{\zfr\zfr}$. For $u\in P_{\ell}$, \begin{align} [u,q_{\zfr\zfr}]&=0,\label{eq:zz}\ [u,q_{\sfr\zfr}]&=\sum_{\alpha}\lambda_{\ell}(s_{\alpha})z_{\alpha}\wedge J_{\ell}u, \qquad q_{\sfr\zfr}=\sum_{\alpha}s_{\alpha}\wedge z_{\alpha},\label{eq:sz}\ [u,q_{\sfr\sfr}]&=K_{\ell}(q_{\sfr\sfr})\wedge J_{\ell}u, \qquad K_{\ell}(q_{\sfr\sfr})\in\sfr.\label{eq:ss} \end{align} The $\wedge^2\zfr$ component is zero, the $\wedge^2\sfr$ component lies in the noncentral rotated sector removed by the normal-form projection, and the $\sfr\wedge\zfr$ component is the only component capable of changing the central rotated representative. Its effect on $[\sigma_R]$ is a $d_{\ufr}$-coboundary. \end{lemma}

\begin{proof} Equations \eqref{eq:zz}--\eqref{eq:ss} follow from $[u,z]=0$ and $[u,s]=-\lambda_{\ell}(s)J_{\ell}u$. If $r$ is replaced by $r+q$ and $R$ by $R-d_{\g}q$, the $\sfr\wedge\zfr$ component changes the representative by

$$\Delta\sigma(u)=J_{\ell}u\wedge\sum_{\alpha}\lambda_{\ell}(s_{\alpha})R(z_{\alpha}).$$

With $\Theta=-\sum_{\alpha}s_{\alpha}\wedge R(z_{\alpha})$, one has $d_{\ufr}\Theta=\Delta\sigma$. Thus the representative may change, but the Hochschild--Serre class does not. If one fixes the normal-form convention in which $W_{\ell}$ is represented modulo this $\sfr\wedge\zfr$ coboundary freedom, then $W_{\ell}$ itself is unchanged in the quotient. \end{proof}

\begin{theorem}[Canonicity]\label{thm:canonicity} The class $[\sigma_R]\in E_{2,J}^{0,1}\cong\coker\beta$ depends only on the cohomology class of the cobracket $\xi=\ad_r+R$ and on the Milnor decomposition. It is independent of the choice of $r$, of oriented orthonormal bases of the planes $P_{\ell}$, and of the normal-form complement $\calH$. \end{theorem}

\begin{proof} Changes of $r$ by elements of $\wedge^2\afr$ are separated by \Cref{lem:projectors}; all three components act trivially on the class. Orientation-preserving basis changes commute with $J_{\ell}$ and fix $\omega_{\ell}$. Orientation reversal changes $(\lambda_{\ell},J_{\ell},W_{\ell})$ to $(-\lambda_{\ell},-J_{\ell},-W_{\ell})$, leaving $J_{\ell}u\wedge R(W_{\ell})$ invariant. Finally, two choices of $\calH$ differ by a finite-dimensional coboundary from $\wedge^2\afr$, already covered by the projector argument. \end{proof}

\section{Polynomial integration}\label{sec:integration}

Let $N=\exp(\zfr\oplus\ufr)$. This group is abelian, but the coefficient action of $\zfr\oplus\ufr$ on $\wedge^2\g$ is nontrivial through the bracket with $\sfr$.

\begin{theorem}[Polynomial multiplicative integration]\label{thm:integration} Let $R\in Z^1(\g,\wedge^2\g)$. Let $\Pi_R^R$ be the right-trivialized multiplicative Poisson tensor integrating $R|_{\zfr\oplus\ufr}$:

$$\Pi_R^R(n)=(R_{n^{-1}})_{\ast}\Pi_R(n).$$

For $X\in\zfr\oplus\ufr$, \begin{equation}\label{eq:integration} \Pi_R^R(\exp X)=\int_0^1 e^{t\ad_X}R(X),dt =R(X)+\frac12[X,R(X)]. \end{equation} Thus the coefficients in exponential coordinates on $N$ are polynomials of degree at most two. \end{theorem}

\begin{proof} For a Lie algebra one-cocycle, the right-trivialized group cocycle satisfies

$$\frac{d}{dt}\Pi_R^R(\exp(tX))=e^{t\ad_X}R(X), \qquad \Pi_R^R(e)=0.$$

Integration gives the first equality in \eqref{eq:integration}. Since $\zfr\oplus\ufr$ is abelian, BCH on $N$ truncates exactly:

$$\exp X\exp Y=\exp(X+Y).$$

Furthermore

$$[X,\zfr]=0, \qquad [X,\ufr]=0, \qquad [X,\sfr]\subset\ufr,$$

so $\ad_X^2=0$ on $\g$ and hence on $\wedge^2\g$. Therefore $e^{t\ad_X}=\id+t\ad_X$ exactly. Since $R(X)$ is linear in $X$, the two terms in \eqref{eq:integration} are respectively linear and quadratic. \end{proof}

\begin{remark}[Necessity of the abelian nilradical] If the nilradical is nonabelian, the degree bound need not hold. For a nilpotent ideal of step $c$, BCH and $e^{t\ad_X}$ can produce higher finite-degree terms; for a nonnilpotent normal subgroup one may obtain nonpolynomial exponential or trigonometric coefficients. The degree $\leq2$ assertion is therefore specific to Milnor's condition $[\ufr,\ufr]=0$. \end{remark}

\section{Resonant singular perturbations}\label{sec:resonance}

Fix a smooth one-parameter family of flat metric Lie algebras with weights

$$\lambda_{\ell}^{\varepsilon}=\lambda_{\ell}+\varepsilon\nu_{\ell},$$

nonresonant for $\varepsilon>0$, and with a single transversal resonance \begin{equation}\label{eq:transverse} \lambda_i+\sigma\lambda_j=0, \qquad \nu_i+\sigma\nu_j\neq0. \end{equation} The mixed resonant plane $\calM_{ij}^{\sigma}\subset P_i\wedge P_j$ is the two-dimensional weight space whose weight is $\varepsilon(\nu_i+ \sigma\nu_j)$.

All analytic statements below are finite-dimensional. We identify the underlying vector spaces of $\g_{\varepsilon}$ with that of $\g_0$; smooth dependence means smooth dependence of structure constants and Milnor projectors. Laurent expansions are taken in a punctured real interval, equivalently after complexification in a punctured disc. All norms are fixed Euclidean norms, hence equivalent.

For $q\in\calM_{ij}^{\sigma}$ define \begin{equation}\label{eq:residue} \calC_{\nu}(q)(s)=(\nu_i+\sigma\nu_j)(s)K_{ij}^{\sigma}q, \quad s\in\sfr, \qquad \calC_{\nu}(q)|_{\zfr\oplus\ufr}=0, \end{equation} where $K_{ij}^{\sigma}$ is the mixed complex structure on $\calM_{ij}^{\sigma}$.

\begin{proposition}[Residue cocycle]\label{prop:residue} The map $\calC_{\nu}(q)$ is a Chevalley--Eilenberg one-cocycle in $Z^1(\g_0,\wedge^2\g_0)$, and

$$\ad_{\varepsilon}(\varepsilon^{-1}q)\to\calC_{\nu}(q) \qquad (\varepsilon\to0).$$

\end{proposition}

\begin{proof} On $\calM_{ij}^{\sigma}$ the zeroth-order weight cancels and $[s,q]_{\varepsilon}=\varepsilon(\nu_i+ \sigma\nu_j)(s)K_{ij}^{\sigma}q$. Dividing by $\varepsilon$ gives \eqref{eq:residue}. Since the limit has weight zero on the resonant fibre and vanishes on the abelian ideal $\zfr\oplus\ufr$, the Chevalley--Eilenberg cocycle identity is immediate. \end{proof}

\begin{theorem}[Simple-pole theorem]\label{thm:simplepole} Let $r_{\varepsilon}$ be the Milnor-normal solution of the reduced nonresonant homological equation

$$D_{\varepsilon}r_{\varepsilon}=\Psi_{\varepsilon}, \qquad \Psi_{\varepsilon}\in(\wedge^3\g)^{\g_{\varepsilon}},$$

with $\Psi_{\varepsilon}$ uniformly bounded and with only the transversal resonance \eqref{eq:transverse}. Then

$$r_{\varepsilon}=\varepsilon^{-1}q+r_0+o(1), \qquad q\in\calM_{ij}^{\sigma},$$

and no pole of order $\ge2$ can occur. The residue $q$ determines the cocycle $\calC_{\nu}(q)$ of \eqref{eq:residue}. \end{theorem}

\begin{proof} On nonresonant weight blocks the inverse of the weight operator is uniformly bounded. On $\calM_{ij}^{\sigma}$ the relevant eigenvalue is $\varepsilon(\nu_i+ \sigma\nu_j)$ times an invertible complex structure, so the inverse has at worst one factor $\varepsilon^{-1}$. If a Laurent expansion had leading order $\varepsilon^{-N}q_N$ with $N\ge2$, the left-hand side $D_{\varepsilon}r_{\varepsilon}$ would contain a term of size $\varepsilon^{1-N}\|q_N\|$ on the resonant block, contradicting boundedness of $\Psi_{\varepsilon}$. Thus $N\le1$, and the leading coefficient lies in the resonant plane. The residue cocycle is \Cref{prop:residue}. \end{proof}

\section{Formal realization at resonance}\label{sec:formal}

Let $\calL$ be the differential graded Lie algebra controlling deformations of the resonant Lie bialgebra $(\g_0,\xi_0)$, with differential

$$L_{\xi_0}(\eta)=\schouten{\mu_0+ \xi_0}{\eta}.$$

Filter $\calL$ by order along the resonant mixed-plane ideal and write $F^k\calL$ for this filtration.

\begin{definition}[Resonant acyclicity]\label{def:acyclic} The resonant fibre is resonantly acyclic if, for every $k\ge1$, \begin{equation}\label{eq:acyclic} H^2(\gr_k^F\calL,L_{\xi_0})=0, \qquad \gr_k^F\calL=F^k\calL/F^{k+1}\calL. \end{equation} Equivalently, every degree-two obstruction with leading resonant order $k$ has a primitive of the same resonant order. \end{definition}

\begin{theorem}[Formal realization criterion]\label{thm:formal} Assume resonant acyclicity. A residue $q\in\calM_{ij}^{\sigma}$ of \Cref{thm:simplepole} is formally realizable by a Laurent-formal solution

$$r(\varepsilon)=\varepsilon^{-1}q+ \sum_{k\ge0}\varepsilon^k r_k, \qquad R(\varepsilon)=\sum_{k\ge0}\varepsilon^kR_k,$$

if and only if \begin{equation}\label{eq:residue-vanishing} [\calC_{\nu}(q)]=0\quad\text{in }H^1(\g_0,\wedge^2\g_0). \end{equation} Under resonant acyclicity this is the complete formal obstruction. \end{theorem}

\begin{proof} Necessity follows by collecting the order-zero part of the Maurer--Cartan equation after subtracting the pole: it is the cocycle $\calC_{\nu}(q)$, which must be a coboundary if a regular formal correction exists. Conversely, choose a primitive killing $\calC_{\nu}(q)$. At order $N$, the remaining defect is $L_{\xi_0}$-closed by the graded Jacobi identity. Its associated graded class lies in $H^2(\gr_N^F\calL,L_{\xi_0})$, which vanishes by \Cref{def:acyclic}; hence it can be killed by an order-$N$ correction. Induction gives the formal solution. \end{proof}

\section{Examples and computations}\label{sec:examples}

\subsection{The Euclidean algebra $\mathfrak e(2)$}

Let $\g=\R s\oplus\R e\oplus\R f$ with $[s,e]=f$, $[s,f]=-e$. Then $\sfr=\R s$, $\zfr=0$, $\ufr=P_1=\R e\oplus\R f$, and $J_1e=f$. Since $\zfr=0$, there is no central rotated coefficient and $\calB_J=0$. Hence $\Ob(R)=0$ for every normalized identity-type structure. The polynomial integration formula gives, for $u=xe+yf$ and $R(e)=s\wedge e$, $R(f)=s\wedge f$,

$$\Pi_R^R(\exp u)=s\wedge u+ \frac12(x^2+y^2)e\wedge f,$$

matching the low-dimensional Poisson--Lie forms in \cite{Stachura1998,Zakrzewski1997}.

\subsection{No nontrivial total four-dimensional obstruction}\label{ssec:fourdim-nogo}

A four-dimensional Milnor algebra with $\ufr\neq0$ has necessarily

$$\dim\sfr=1, \qquad \dim\zfr=1, \qquad m=1.$$

Then $I^2=\R\omega_1$ and $T_1=I^2/\R\omega_1=0$. Therefore $\calB_J=0$ and $\coker\beta=0$. A total four-dimensional example with $\Ob(R)\neq0$ is impossible in the present framework. The minimal explicit obstruction with four-dimensional nilradical is the following example.

\subsection{A worked example with four-dimensional nilradical}\label{ssec:worked}

Let

$$\g=\R s\oplus\R z_1\oplus\R z_2\oplus P_1\oplus P_2, \qquad P_{\ell}=\R e_{\ell}\oplus\R f_{\ell},$$

with $z_1,z_2$ central and

$$[s,e_1]=f_1, \quad [s,f_1]=-e_1, \qquad [s,e_2]=2f_2, \quad [s,f_2]=-2e_2.$$

Thus $\lambda_1(s)=1$, $\lambda_2(s)=2$, and the weights are nonresonant. Set $\zeta=z_1\wedge z_2$ and $\omega_{\ell}=e_{\ell}\wedge f_{\ell}$. Then

$$I^2=\R\zeta\oplus\R\omega_1\oplus\R\omega_2, \quad T_1=\R\zeta\oplus\R\omega_2, \quad T_2=\R\zeta\oplus\R\omega_1.$$

For

$$\Psi=c\,s\wedge\zeta+a_1\wedge\omega_1+a_2\wedge\omega_2, \qquad a_k=x_ks+z_k',$$

\Cref{prop:beta-coordinate} gives \begin{equation}\label{eq:worked-beta} \beta_1(\Psi)=c\zeta+x_2\omega_2, \qquad \beta_2(\Psi)=2c\zeta+2x_1\omega_1. \end{equation} Hence \begin{equation}\label{eq:worked-image} \im\beta={(b_1,b_2):\text{the $\zeta$-coefficient of }b_2=2\text{ times that of }b_1}. \end{equation}

\begin{center} \renewcommand{\arraystretch}{1.16} \begin{tabular}{c c c c c c c} \hline plane & basis & $\lambda_{\ell}(s)$ & $J_{\ell}$ & $A_{\ell}$ & $W_{\ell}$ & target \ \hline $P_1$ & $(e_1,f_1)$ & $1$ & $J_1e_1=f_1$ & $0$ & $z_1$ & $T_1=\R\zeta\oplus\R\omega_2$ \ $P_2$ & $(e_2,f_2)$ & $2$ & $J_2e_2=f_2$ & $0$ & $z_2$ & $T_2=\R\zeta\oplus\R\omega_1$ \ \hline \end{tabular} \end{center}

Define \begin{align*} R(e_1)&=z_1\wedge f_1,& R(f_1)&=-z_1\wedge e_1,\ R(e_2)&=z_2\wedge f_2,& R(f_2)&=-z_2\wedge e_2, \end{align*} with $R(s)=0$ and with central values in $I^2$ as below. These formulas define a one-cocycle because the central values are invariant and the $s$-action on $P_{\ell}$ is by $\lambda_{\ell}J_{\ell}$.

\subsubsection*{Obstructed case} Let

$$R(z_1)=0, \qquad R(z_2)=\zeta.$$

Then

$$(\pi_1R(W_1),\pi_2R(W_2))=(0,\zeta).$$

By \eqref{eq:worked-image}, this tuple is not in $\im\beta$, because $1\neq2\cdot0$. Thus

$$\Ob(R)=[(0,\zeta)]\neq0.$$

The representative is

$$\sigma_R(e_1)=\sigma_R(f_1)=0, \qquad \sigma_R(e_2)=f_2\wedge\zeta, \qquad \sigma_R(f_2)=-e_2\wedge\zeta.$$

By \Cref{thm:obstruction}, the coupled Maurer--Cartan system has no solution.

\subsubsection*{Resolved case} Let

$$R(z_1)=\zeta, \qquad R(z_2)=2\zeta.$$

Then

$$(\pi_1R(W_1),\pi_2R(W_2))=(\zeta,2\zeta)=\beta(s\wedge\zeta).$$

Consequently $\Ob(R)=0$. The cochain

$$\sigma_R(e_1)=f_1\wedge\zeta, \quad \sigma_R(f_1)=-e_1\wedge\zeta, \quad \sigma_R(e_2)=2f_2\wedge\zeta, \quad \sigma_R(f_2)=-2e_2\wedge\zeta$$

is $d_{\ufr}(-s\wedge\zeta)$, so $[\sigma_R]=0$ and the $J$-channel equation is solved by $\Psi=s\wedge\zeta$.

\subsection{Algorithmic computation}

The obstruction is computed as follows: list the weights and central data; form $I^2$ and each $T_{\ell}$; build the matrix of $\beta$ from \eqref{eq:beta-coordinate}; assemble the vector $(\pi_{\ell}R(W_{\ell}))_{\ell}$; reduce it modulo the column space of $\beta$. The reduced vector is $\Ob(R)$.

\section{Conclusion}

The central rotated $J$-channel adds a genuine finite-dimensional obstruction to the identity-type theory of flat metric Lie bialgebras. The obstruction is simultaneously a concrete cokernel class and a canonical Hochschild--Serre edge class. Strict cochain-level invariance guarantees spectral survival, while cohomological invariance alone requires the transgression test. The rank formulas show when the obstruction is forced to vanish and when it can appear. The examples demonstrate that nonzero obstruction is impossible in total dimension four but appears naturally once the visible central and cross-plane targets outnumber the image of the weight contraction map.

\section*{Declarations} \noindent\textbf{Data availability.} This is a theoretical work and no data sets were generated or analyzed.

\noindent\textbf{Competing interests.} The author declares no competing interests.

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